Working Paper: NBER ID: w29879
Authors: Clément de Chaisemartin
Abstract: I consider estimation of the average treatment effect (ATE), in a population composed of $G$ groups, when one has unbiased and uncorrelated estimators of each group's conditional average treatment effect (CATE). These conditions are met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by B standard deviations of the outcome, for some known B. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This minimax-linear estimator assigns a weight equal to group g's share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. I also derive the minimax-linear estimator when the CATEs' estimators are positively correlated, a condition that may be met by differences-in-differences estimators in staggered adoption designs.
Keywords: bias-variance tradeoff; average treatment effect; mean-squared error; minimax linear estimator; bounded normal mean model; stratified randomized experiments; differences-in-differences; staggered adoption designs; shrinkage
JEL Codes: C21; C23
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| CATEs positively correlated (C10) | minimax linear estimator still feasible (C51) |
| CATEs (Y90) | ATE (Y60) |
| CATEs (unbiased and uncorrelated) (C46) | minimax linear estimator (C51) |
| minimax linear estimator (C51) | lower variance (C29) |